For questions about recurrence relations, convergence tests, and identifying sequences

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19
votes
4answers
2k views

What's the value of $\sum\limits_{k=1}^{\infty}\frac{k^2}{k!}$?

For some series, it is easy to say whether it is convergent or not by the "convergence test", e.g., ratio test. However, it is nontrivial to calculate the value of the sum when the series converges. ...
13
votes
4answers
722 views

Slowing down divergence 2

Let $f(x)$ and $g(x)$ be positive nondecreasing functions such that $ \sum_{n>1} \frac1{f(n)} \text{ and } \sum_{n>1} \frac1{g(n)} $ diverges. (Why) must the series $$\sum_{n>1} ...
25
votes
2answers
5k views

$\sum k! = 1! +2! +3! + \cdots + n!$ ,is there a generic formula for this?

I came across a question where I needed to find the sum of the factorials of the first $n$ numbers. So I was wondering if there is any generic formula for this? Like there is a generic formula for ...
12
votes
4answers
566 views

Do these series converge to logarithms?

It is well known that $$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}... =\log(2).$$ If we consider the array: $T(n,k) = -(n-1)\; \text{ if }\; n|k, \;\text{ else } \;1,$ Starting: ...
11
votes
1answer
484 views

Convergence of $\sum_{n=1}^\infty \frac{\sin^2(n)}{n}$

Does the series $$ \sum_{n=1}^\infty \frac{\sin^2(n)}{n} $$ converge? I've tried to apply some tests, and I don't know how to bound the general term, so I must have missed something. Thanks in ...
10
votes
2answers
505 views

How to find $\zeta(0)=\frac{-1}{2}$ by definition?

I would like to know how we can find the following result: $$\zeta(0)=-\frac12$$ Is there a way, using the definition, $$\zeta(s)=\sum_{i=1}^{\infty}i^{-s}$$ to find this?
6
votes
3answers
536 views

Summing the series $(-1)^k \frac{(2k)!!}{(2k+1)!!} a^{2k+1}$

How does one sum the series, $$ S = a -\frac{2}{3}a^{3} + \frac{2 \cdot 4}{3 \cdot 5} a^{5} - \frac{ 2 \cdot 4 \cdot 6}{ 3 \cdot 5 \cdot 7}a^{7} + \cdots $$ This was asked to me by a high school ...
4
votes
4answers
5k views

How does the sum of the series “$1 + 2 + 3 + 4 + 5 + 6\ldots$” to infinity = “$-1/12$”? [duplicate]

(I was requested to edit the question to explain why it is different that a proposed duplicate question. This seems counterproductive to do here, inside the question it self, but that is what I have ...
20
votes
3answers
1k views

Generalizing $\sum \limits_{n=1}^{\infty }n^{2}/x^{n}$ to $\sum \limits_{n=1}^{\infty }n^{p}/x^{n}$

For computing the present worth of an infinite sequence of equally spaced payments $(n^{2})$ I had the need to evaluate ...
15
votes
1answer
696 views

Is there a constructive proof of this characterization of $\ell^2$?

I would like to revisit this question, which can be equivalently stated as: Proposition. Let $(a_n)$ be a sequence of real (or complex) numbers such that $\sum a_n b_n$ converges for every $(b_n) ...
14
votes
3answers
574 views

is a net stronger than a transfinite sequence for characterizing topology?

For metric spaces, knowledge of the convergence of sequences determines the topology completely. A set is closed in the metric topology if and only if it is closed under the limit of convergent ...
7
votes
8answers
3k views

I have to show $(1+\frac1n)^n$ is monotonically increasing sequence

I have to show $(1+\frac1n)^n$ is monotonically increasing sequence. $U_n=(1+\frac1n)^n$, $U_{n+1}=(1+\frac1{n+1})^{n+1}$ I must show $U_{n+1}-U_n\geq0$ i.e. to show ...
5
votes
3answers
298 views

Does $\sum _{k=2} ^\infty \frac{(-1)^k}{\sqrt{k}+(-1)^k}$ converge conditionally?

Discuss the convergence of the sum: $$\sum _{k=2} ^\infty \frac{(-1)^k}{\sqrt{k}+(-1)^k}.$$ My answer so far: It does not converge absolutely since $$\left | \frac{(-1)^k}{\sqrt{k}+(-1)^k} \right ...
3
votes
6answers
1k views

Bernoulli's representation of Euler's number, i.e $e=\lim \limits_{x\to \infty} \left(1+\frac{1}{x}\right)^x $ [duplicate]

Possible Duplicates: Finding the limit of $n/\sqrt[n]{n!}$ How come such different methods result in the same number, $e$? I've seen this formula several thousand times: $$e=\lim_{x\to ...
6
votes
3answers
5k views

Summation involving a factorial: $1 + \sum_{j=1}^{n} j!j$ [duplicate]

$$1 + \sum_{j=1}^{n} j!j$$ I want to find a formula for the above and then prove it by induction. The answer according to Wolfram is $(n+1)!-1$, however I have no idea how to get there. Any hints or ...
2
votes
3answers
299 views

Minimal solution to the sum using median

Suppose we have a set $S$ of real numbers. Show that $$\sum_{s\in S}|s-x| $$ is minimal if $x$ is equal to the median. This is a sample exam question of one of the exams that I need to take and I ...
7
votes
5answers
586 views

Proving $\sum_{k=1}^n k\cdot k! = (n+1)!-1$ without using mathematical Induction. [duplicate]

Possible Duplicate: Summation of a factorial This equation is given: $$ 1\cdot1! + 2\cdot2! + 3\cdot3! + \ldots + n\cdot n! = (n+1)! - 1 $$ I've solved it using mathematical induction but ...
5
votes
2answers
254 views

Convergence of $\sum_{n=2}^\infty \frac{1}{n^\alpha \ln^\beta (n)} $

Study the convergence of the following series: $$\sum_{n=2}^\infty \frac{1}{n^\alpha \cdot\ln^\beta(n)} \text{ where }\alpha,\beta \geq 0 $$ Applying d'Alembert criterion I have that $$ ...
3
votes
4answers
8k views

Is the sum of all natural numbers $-\frac{1}{12}$? [duplicate]

My friend showed me this youtube video in which the speakers present a line of reasoning as to why $$ \sum_{n=1}^\infty n = -\frac{1}{12} $$ My reasoning, however, tells me that the previous ...
2
votes
3answers
263 views

Calculating $1+\frac13+\frac{1\cdot3}{3\cdot6}+\frac{1\cdot3\cdot5}{3\cdot6\cdot9}+\frac{1\cdot3\cdot5\cdot7}{3\cdot6\cdot9\cdot12}+\dots? $

How to find infinite sum How to find infinite sum $$1+\dfrac13+\dfrac{1\cdot3}{3\cdot6}+\dfrac{1\cdot3\cdot5}{3\cdot6\cdot9}+\dfrac{1\cdot3\cdot5\cdot7}{3\cdot6\cdot9\cdot12}+\dots? $$ I can see ...
5
votes
2answers
576 views

What's the limit of the sequence $\lim_{n \rightarrow \infty} \frac{n!}{n^n}$?

$\displaystyle \lim_{n \rightarrow \infty} \frac{n!}{n^n}$ I have a question: Is it valid to use Stirling's Formula to prove convergence of the sequence?
1
vote
2answers
797 views

Proving $\sum\limits_{i=0}^n i 2^{i-1} = (n+1) 2^n - 1$ by induction

I'm trying to apply an induction proof to show that $((n+1) 2^n - 1 $ is the sum of $(i 2^{i-1})$ from $0$ to $n$. the base case: L.H.S = R.H.S we assume that $(k+1) 2^k - 1 $ is true. we need to ...
75
votes
4answers
2k views

Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$

I would like to prove that $\displaystyle\sum_{\substack{n=1\\n\text{ odd}}}^{\infty}\frac{n}{e^{n\pi}+1}=\frac1{24}$. I found a solution by myself 10 hours after I posted it, here it is: ...
34
votes
3answers
1k views

Does the series $ \sum\limits_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}} $ converge or diverge?

Does the following series converge or diverge? I would like to see a demonstration. $$ \sum_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}}. $$ I can see that: $$ \sum_{n=1}^{\infty} \frac{1}{n^{1 + ...
34
votes
5answers
1k views

A question on Taylor Series and polynomial

Suppose $ f(x)$ that is infinitely differentiable in $[a,b]$. For every $c\in[a,b] $ the series $\sum\limits_{n=0}^\infty \cfrac{f^{(n)}(c)}{n!}(x-c)^n $ is a polynomial. Is true that $f(x)$ is a ...
23
votes
3answers
425 views

$\sum_{n=1}^{\infty}\frac{H_n}{n^2 2^n}=\zeta(3)-\frac{1}{2}\log(2)\zeta(2)$

How can I prove that $$\sum_{n=1}^{\infty}\frac{H_n}{n^2 2^n}=\zeta(3)-\frac{1}{2}\log(2)\zeta(2).$$ Can anyone help me please?
19
votes
4answers
514 views

A sum containing harmonic numbers $\sum_{n=1}^\infty\frac{H_n}{n^3\,2^n}$

I'm trying to find a closed form for the following sum $$\sum_{n=1}^\infty\frac{H_n}{n^3\,2^n},$$ where $H_n=\displaystyle\sum_{k=1}^n\frac{1}{k}$ is a harmonic number. Could you help me with it?
18
votes
2answers
2k views

A hard definite integral with trigonometric functions

How could we get a closed form for this one? $$\displaystyle\int_{0}^{\frac{\pi }{2}}{{{x}^{2}}\sqrt{\tan x}\sin \left( 2x \right)\text{d}x}$$
30
votes
3answers
1k views

Prove the convergence/divergence of $\sum \limits_{k=1}^{\infty} \frac{\tan(k)}{k}$

Can be easily proved that the following series onverges/diverges? $$\sum_{k=1}^{\infty} \frac{\tan(k)}{k}$$ I'd really appreciate your support on this problem. I'm looking for some easy proof here. ...
41
votes
5answers
3k views

$n$th derivative of $e^{1/x}$

I am trying to find the $n$'th derivative of $f(x)=e^{1/x}$. When looking at the first few derivatives I noticed a pattern and eventually found the following formula $$\frac{\mathrm d^n}{\mathrm ...
27
votes
3answers
1k views

Why is this series of square root of twos equal $\pi$?

Wikipedia claims this but only cites an offline proof: $$\lim_{n\to\infty} 2^n \sqrt{2-\sqrt{2+... \sqrt 2}} = \pi$$ for $n$ square roots and one minus sign. The formula is not the "usual" one, like ...
19
votes
4answers
4k views

Convergence of the sequence $\sqrt{1+2\sqrt{1}},\sqrt{1+2\sqrt{1+3\sqrt{1}}},\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1}}}},\cdots$

I recently came across this problem Q1 Show that $\lim\limits_{n \rightarrow \infty} \underbrace{{\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots+n\sqrt{1}}}}}}}_{n \textrm{ times }} = 3$ After ...
15
votes
2answers
484 views

Closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$.

What is the closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$? We can use Fourier series of $e^{-bx}$ ($|x|<\pi$) to evaluate $\sum_{n=-\infty}^{\infty}\frac{1}{n^2+b^2}$. But this ...
22
votes
2answers
817 views

Find the exact value of the infinite sum $\displaystyle\sum_{n=1}^\infty \left\{\mathrm{e}-\Big(1+\frac1n\!\Big)^{n}\right\}$

How can we find the exact value of the infinite sum $$ \displaystyle\sum_{n=1}^\infty \left\{\mathrm{e}-\Big(1+\frac1n\Big)^n\right\}? $$ This problem appears in: T. Andreescu, T. Radulescu & V. ...
17
votes
4answers
1k views

The sum of $(-1)^n \frac{\ln n}{n}$

I'm stuck trying to show that $$\sum_{n=2}^{\infty} (-1)^n \frac{\ln n}{n}=\gamma \ln 2- \frac{1}{2}(\ln 2)^2$$ This is a problem in Calculus by Simmons. It's in the end of chapter review and it's ...
12
votes
4answers
918 views

Testing the series $\sum\limits_{n=1}^{\infty} \frac{1}{n^{k + \cos{n}}}$

We know that the "Harmonic Series" $$ \sum \frac{1}{n}$$ diverges. And for $p >1$ we have the result that the series converges $$\sum \frac{1}{n^{p}}$$ converges. One can then ask the question of ...
13
votes
2answers
916 views

Evaluating $\sum_{n=1}^{\infty} \frac{n}{e^{2 \pi n}-1}$ using the inverse Mellin transform

Inspired by this post, I'm trying to evaluate $\displaystyle \sum_{n=1}^{\infty} \frac{n}{e^{2 \pi n}-1} = \frac{1}{24} - \frac{1}{8 \pi}$ using the inverse Mellin transform. But my answer is twice ...
9
votes
2answers
1k views

Does $\sum \limits_{n=1}^\infty\frac{\sin n}{n}(1+\frac{1}{2}+\cdots+\frac{1}{n})$ converge (absolutely)?

I've had no luck with this one. None of the convergence tests pop into mind. I tried looking at it in this form $\sum \sin n\frac{1+\frac{1}{2}+\cdots+\frac{1}{n}}{n}$ and apply Dirichlets test. I ...
8
votes
5answers
366 views

Finding sum of a series: difference of cubes

I am trying to find sum of the infinite series: $$1^{3}-2^{3}+3^{3}-4^{3}+5^{3}-6^{3} + \ldots$$ I tried to solve it by subtracting sum of even cubes from odd, but that solves only half of the ...
7
votes
5answers
1k views

Prove $\lim_{x \rightarrow 0} \frac {\sin(x)}{x} = 1$ with the epsilon-delta definition of limit.

It is well known that $$\lim_{x \rightarrow 0} \frac {\sin(x)}{x} = 1$$ I know several proofs of this: the geometric proof shows that $\cos(\theta)\leq\frac {\sin(\theta)}{\theta}\leq1$ and using ...
6
votes
3answers
355 views

sum of arctangent

Here is an interesting topic. It comes from evaluating $$\sum_{k=1}^{\infty}tan^{-1}\left(\frac{1}{k^{2}}\right)$$ I managed to dig up an old paper I have on the sum of arctans by Boros and Moll. ...
8
votes
4answers
616 views

Find the limit $ \lim_{n \to \infty}\left(\frac{a^{1/n}+b^{1/n}+c^{1/n}}{3}\right)^n$

For $a,b,c>0$, Find $$ \lim_{n \to \infty}\left(\frac{a^{1/n}+b^{1/n}+c^{1/n}}{3}\right)^n$$ how can I find the limit of sequence above? Provide me a hint or full solution. thanks ^^
10
votes
1answer
501 views

Proving that $ \frac{1}{\sin(45°)\sin(46°)}+\frac{1}{\sin(47°)\sin(48°)}+…+\frac{1}{\sin(133°)\sin(134°)}=\frac{1}{\sin(1°)}$

I would like to show that the following trigonometric sum $$ \frac{1}{\sin(45°)\sin(46°)}+\frac{1}{\sin(47°)\sin(48°)}+\cdots+\frac{1}{\sin(133°)\sin(134°)}$$ ...
8
votes
4answers
381 views

Find the infinite sum of the series $\sum_{n=1}^\infty \frac{1}{n^2 +1}$

This is a homework question whereby I am supposed to evaluate: $$\sum_{n=1}^\infty \frac{1}{n^2 +1}$$ Wolfram Alpha outputs the answer as $$\frac{1}{2}(\pi \coth(\pi) - 1)$$ But I have no idea ...
7
votes
5answers
795 views

Need to prove the sequence $a_n=1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2}$ converges

I need to prove that the sequence $a_n=1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2}$ converges. I do not have to find the limit. I have tried to prove it by proving that the sequence is monotone ...
15
votes
4answers
553 views

Generalisation of the identity $\sum\limits_{k=1}^n {k^3} = \bigg(\sum\limits_{k=1}^n k\bigg)^2$

Are there any generalisations of the identity $\sum\limits_{k=1}^n {k^3} = \bigg(\sum\limits_{k=1}^n k\bigg)^2$ ? For example can $\sum {k^m} = \left(\sum k\right)^n$ be valid for anything other than ...
13
votes
6answers
3k views

Prove the divergence of the sequence $(\sin(n))_{n=1}^\infty$.

I am looking for nice ways of proving the divergence of the sequence $(x_n)_{n=1}^\infty$ defined by $$x_n:=\sin(n).$$ One (not so nice) way is to construct two subsequences: one where the indexes are ...
10
votes
4answers
190 views

Convergence of $\frac{\sqrt{a_{n}}}{n}$

Can anyone help me with the following question. If $a_{n} \geq 0$ and $\sum a_{n}$ converges then how to prove $\sum \frac{\sqrt{a_{n}}}{n}$ converges. Any idea where to start. My idea was to try ...
8
votes
3answers
316 views

Value of $\sum_{k=1}^{\infty}\frac{1}{k^2+a^2}$

So my question is to find the value of $\sum_{k=0}^\infty\frac{1}{k^2+1}$ and more generally $\sum_{k=0}^\infty\frac{1}{Q(k)}$ where Q is a quadratic polynomial with no zeroes on the integers. I can ...
7
votes
1answer
288 views

Convergence of series involving in iterated logarithms $\sum \frac{1}{n(\log n)^{\alpha_1}\cdots (\log^k(n))^{\alpha_k} }$

What is the quickest way to show when $$ S(\alpha_1,\alpha_2,\cdots,\alpha_k) = \sum\limits_{n=3}^\infty \frac{1}{n (\log n)^{\alpha_1}\cdots (\log^k(n))^{\alpha_k}} $$ converges, where $\log^k(n)$ ...